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Measurement and analysis of machine tool errors under quasi-static and loaded conditions
Accepted Manuscript Title: Measurement and analysis of machine tool errors under quasi-static and loaded conditions Authors: K´aroly Szipka, Theodoros Laspas, Andreas Archenti PII: DOI: Reference:
S0141-6359(17)30201-5 http://dx.doi.org/doi:10.1016/j.precisioneng.2017.07.011 PRE 6621
To appear in:
Precision Engineering
Received date: Revised date: Accepted date:
27-4-2017 20-7-2017 25-7-2017
Please cite this article as: Szipka K´aroly, Laspas Theodoros, Archenti Andreas.Measurement and analysis of machine tool errors under quasi-static and loaded conditions.Precision Engineering http://dx.doi.org/10.1016/j.precisioneng.2017.07.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Measurement and analysis of machine tool errors under quasi-static and loaded conditions Károly Szipka*, Theodoros Laspas, Andreas Archenti KTH Royal Institute of Technology, Brinellvägen 68, 10044 Stockholm * Corresponding author. Tel.: +46-736-737-825 . E-mail address: [email protected]
Highlights
A novel methodology for the prediction of machine tool errors under quasi-static and loaded conditions Measurement and analysis of the combined effect of geometric errors and quasi-static load induced deviations on machine tools Estimation of deflections through the determination of the position- and direction-dependent static stiffness of machine tools Introduction how the static stiffness characteristic of machine tools is mapped on the part geometry
Abstract Machine tool testing and accuracy analysis has become increasingly important over the years as it offers machine tool manufacturers and end-users updated information on a machine’s capability. A machine tool´s capability may be determined by mapping the distribution of deformations and their variation range, in the machine tool workspace, under the cumulative effect of thermal and mechanical loads. This paper proposes a novel procedure for the prediction of machine tool errors under quasi-static and loaded conditions. Geometric errors and spatial variation of static stiffness in the work volume of machines are captured and described through the synthesis of bottom-up and top-down model building approaches. The bottom-up approach, determining individual axis errors using direct measurements, is applied to estimate the geometric errors in unloaded condition utilizing homogeneous transformation matrix theory. The top-down approach, capturing aggregated quasi-static deviations using indirect measurements, estimates through an analytical procedure the resultant deviations under loaded conditions. The study introduces a characterization of the position and direction dependent static stiffness and presents the identification how the quasi-static behavior of the machine tool affects the part accuracy. The methodology was implemented in a case study, identifying a variation of up to 27% in the stiffness response of the machine tool. The prediction results were experimentally validated through cutting tests and the uncertainty of the measurements and the applied methodology was investigated to determine the reliability of the predicted errors. Keywords: Machine tool, accuracy, quasi-static stiffness, geometric error
1. Introduction Thorough characterization of machine tools’ accuracy requires quasi-static analysis under loaded and unloaded conditions. Machine tool accuracy analysis can be used to extract valuable information through identification of the separate effects of the various error sources, including kinematic, static, thermal, dynamic, and motion control [1]. For machine tool geometric error analysis, several different schemes have been proposed for ‘direct’ [2] and ‘indirect’ [3] measurements. Direct methods are implemented through error motion measurements of a single axis while indirect methods require simultaneous motion of two or more axis of the machine under test for estimating overall geometric errors. Analysis of superposed errors for a chain of axis uses a ‘‘bottom-up’’ approach, based on Hartenberg–Denavit transformation [4], where motion errors of individual components are added to the aggregated volumetric error [5]. The indirectly measured aggregated errors [6] can be analyzed through a “top-down” approach to derive the deviation matrix of the system. ISO 230-1 [7] specifies methods and guidelines for testing the accuracy of machine tools under unloaded or quasi-static conditions accounting only for geometric errors and quasi-static load induced deviations. Under machining conditions, however,
the induced deviations due to loads have a dependency on the machine tool stiffness and influence the system’s precision. Machine tool stiffness represents its ability to resist deflections due to e.g. cutting forces and have a significant effect on the quasistatic and dynamic accuracy. Load induced deformations have been commonly described and investigated by various types of finite element (FE) approaches [8] or combination of FE and analytical methods [9] rather than measurement based modeling approaches [10], [11]. [12] and in dynamic case [13] presents direction dependent stiffness investigations. The main limitation with the existing methods is the lack of appropriate measurement approaches that link individual axis errors under the influence of loads to the aggregated machine tool accuracy. The reasons can partly be explained by the lack of proper instrumentation, which is usually limited to the measurement of displacement in one position and direction for one setup [7], neglecting the characterization of the machine tool`s response in the whole workspace.
Fig. 1. The synthesis of bottom-up and top down modeling approach
The proposed methodology relies on the synthesis of bottom-up (geometric errors stack-up) and top-down (position dependent variation of static stiffness) approaches and enables the relaying of the investigated error sources through the combination of indirect and direct measurements (Fig. 1). Hence, the two approaches are acting complementary to each other covering different aspects of the machine tool performance with enabling the possibility for assessment of the pure machine tool deflections without the geometric errors. The core novelty of this work relies on the top-down approach and the synthesis of the two approaches. The bottom-up approach aims in evaluating the geometric accuracy of the machine tool under unloaded conditions at the trajectory of the functional point (as described in ISO 230-1 [7]). Using homogeneous transformation matrices (HTM) the geometric accuracy of the machine tool is determined by superposing the geometric errors of each individual machine axis. Direct measurement of the errors can be carried out using a laser interferometer or other standard measurement methods [2]. The top-down approach aims at mapping the static stiffness response of the machine tool by measuring the aggregated deviations under loaded conditions and hence allows the integration of small machine structure deformations in the geometric error model. In addition, it enables the identification of dependencies of stiffness variation in the workspace due to axes positions and applied force direction. An indirect measurement procedure for measuring the aggregated errors is realized experimentally by the test instrument Loaded Double Ball Bar (LDBB) [14], which is capable of applying forces of controlled magnitude and orientation and measure the induced deviations identifying the stiffness variation under different machine spatial configurations. The LDBB is seen as an elastic link – as described by the Elastic Linked System (ELS) [15] framework – that closes the force loop between the tool and the workpiece and represents the static response of the machine structure to the cutting process. In section 2 of the paper an introduction of the methodology for the assessment of geometric and quasi-static load dependent errors is provided with a focus on the top-down approach. Section 3 covers the implementation of the proposed methodology, with additional measurement results in the appendix, and Section 4 presents the experimental validation procedure. Section 5 focuses on the measurement uncertainties of both of the approaches before conclusions are discussed in Section 6. 2. Methodology for the assessment of geometric and quasi-static load dependent errors 2.1. The bottom-up approach Characterization of individual axis position-dependent and position-independent geometric errors (e.g., linear positioning, straightness, angular or squareness deviation) under unloaded conditions requires a direct measurement method. Laser interferometer systems are commonly used for the task [2] as they enable measurement with high precision and low uncertainty as compared to other methods. In this study a Renishaw XL-80 laser interferometer was used. Since all of the modeling methods which follow the principle of the bottom-up modeling approach (axis geometric error stack-up) are sensitive to the uncertainty of the contributing error components, the uncertainty of the measurements is of importance and will be further analyzed. For a three-axis machine tool 18 position dependent and 3 position independent (location) errors can be identified. Since all zero positions of linear axes can be set to zero when checking the geometric accuracy, any possible offset errors can be neglected. The location errors of the spindle are simplified, which means 4 location errors less to be considered. By selecting Z axis as primary and Y and X as secondary and tertiary axis, the location errors will define the location and orientation with respect to the average line of Z axis. The considered component and location errors can be found in Table 1 using the notation of ISO 230-1 [7]. Table 1. The component and location errors of a three-axis machine tool
Linear errors Axis
Positioning
X axis
EXX
EYX
Y axis
EYY EZZ
EXY EYZ
Z axis Location Errors
Angular errors
Straightness
Roll
Pitch
Yaw
EZX
EAX
EBX
ECX
EZY EXZ
EBY ECZ
EAX EAZ
ECY EBZ
𝐸𝐴0𝑌 squareness error of Y to Z; 𝐸𝐶0𝑋 squareness error of X to Y; 𝐸𝐵0𝑋 squareness error of X to Z;
2.1.1. The computational model (bottom-up) The bottom-up approach is based on HTM theory to describe the kinematic structure of the machine. In order to achieve a comprehensive model, certain steps are defined. A systematic procedure can be outlined through the following steps: 1. Identification of the kinematic structure of the machine model 2. Assignment of the global reference frame and axis local coordinate frames to facilitate the modeling procedure 3. Modeling of the nominal kinematic configuration by describing the relative position and orientation of the axes with respect to each other using HTMs 4. Definition of the measurement functional point where individual error components have been measured 5. Construction of the geometric error model and population with measured data The kinematic structure sets the base on which the entire modeling process will rely, in order to express the motion of the rigid axes and the linking joints. Local and global coordinate frames have to be chosen in order to facilitate the integration of the measurement data with the model. The global coordinate frame is defined to meet the corresponding 0 coordinate for each axis for a three axis machine. An example can be seen in Fig. 2. The model is composed to be applicable for orthogonal axis machine tool with maximum three translational axes. The position independent location errors and the position dependent component errors for each axis give the considered error parameters (see Table 1), which are measured separately. The measured error parameters are coupled into HTMs to express the deviation of each individual axis from the nominal position, the effect on the succeeding rigid body frames and the effect on the accuracy in the position between the tool and workpiece. The HTM theory is implemented for geometric error modeling according to [5] and [16], using the equations defined by these two papers.
Fig. 2. The assignment of the coordinate frames for the three-axis machine tool (AFM R-1000). The coordinate system (O) corresponds to the reference coordinate frame, and coordinate systems (F) correspond to the local coordinate frames of the X, Y and Z axis.
At the end of the process the position and the orientation of the functional point is determined in respect to the table. The relationship between the functional point and the measurement points is a sensitive part of the model as measurements are implemented in one track of the workspace and conclusion is made in another trajectory. To avoid redundancies the lever effect of the angular errors due to the offset between the functional point and the measurement point was compensated. The summarized concept of the bottom-up approach can be seen in Fig. 3.
Machine Tool kinematic structure
Measurement data
Defined toolpath
Statistical analysis
Individual axes position
Geometric error model Composition of HTMs Offset between the measurement and the functional point
Linear and rotational motion carriage with errors of single axes
Estimated geometric errors for specified toolpath Fig. 3. The concept of the bottom-up approach
2.2. The top-down approach The top-down approach is the detailed description of the effect of quasi-static forces, which are related to the static stiffness of the machine tool. In this approach, a computational model is developed to analyze forces and deflections based on an indirect measurement schema, which allows measuring superposed errors of simultaneous motion of two or more machine axes. Investigations were made in one certain plane of the machine tools’ work volume. 2.2.1. Static stiffness of machine tools The stiffness of machine tools generally expresses the capacity of the machine to endure loads within its workspace. The resultant stiffness (aggregated) of a machine tool (at a certain location and direction) is defined through the ratio of the applied load and the induced deformation between the spindle nose and the machine tool table. The loading process has to be slow (quasistatic condition) in case of the static stiffness. For periodic loading this means 1 or 0.5 Hz or lower [8], [17], according to the damping of the mechanical system of the machine tool, to avoid transient movements. The magnitude of the elastic deformations depends on the stiffness of all the components of the machine’s structural chain. In general, it can be stated that the weakest link, the component where the stiffness is the lowest in the stiffness chain, will dominate the response of the whole machine. This means a high level of sensitivity and highlights the importance of stiffness response characterization. Most commonly, the static stiffness of a machine tool is determined through deflection measurements in the directions of the axes, where deflection is measured in line with the applied force. ISO 230-1 [7] introduce the guidelines for the test of machine static compliance at a certain position in the working volume. This method gives an impression about the stiffness state of the machine in the axis direction, but can’t provide enough information for an adequate characterization or an effective compensation. Such measurements can determine a local point stiffness value (kp), which is the magnitude of the applied force (F) divided by the displacement (δ) between the spindle nose with respect to the machine tool table in the load direction: 𝑘𝑝 =
𝐹
𝛿
(1)
Regarding the changing relative position and/or orientation between the mechanical components during motion, the resultant deflections will differ in case of different directions (𝑒𝑓 ) of the force and different positions (p) at the workspace. Consequently the static stiffness of the machine, K p, e f varies as the function of the position and the direction (signum function) of the applied force:
k xx (p,sgn(e f, x )) ... ... K (p,e f ) ... k yy (p,sgn(e f, y )) ... ... ... k zz (p,sgn(e f, z ))
(2)
It has to be noted that certain deflections are the result of internal, assembly dependent loads of the machine, including the effect of the weight of the machine and the preloading of components. Even though these parameters are included in the presented model through the measurements, their change can affect the result. 2.2.2. The mapping of static stiffness characteristic on the part form accuracy It is known that the static stiffness of machine tools varies, both regarding the behavior of each individual machine as well as between different machine tool configurations [15]. This variation, if not compensated, will be mapped on the form accuracy of the part as deviations due to loads. Thus, the static stiffness characteristic of a machine tool is mapped on the form accuracy of the part. The total deviation D (nominal surface) can be decomposed into three different components: deflection of the tool (𝛿 T), deformation of the machine, including fixture and toolholder (𝛿 M), and the deformation of the workpiece (𝛿 W).
𝛿𝐷 = 𝛿𝑇 + 𝛿𝑀 + 𝛿𝑊
(3)
These deviations are the response of the system to the process force. Two characteristics can be identified in the system response: the pulsating component (index p) and a static component (index s). The separation of these two components leads to Eq. 4. A typical example of a measured force in steady-state can be seen in Fig. 4. The static component is responsible for the direct mapping (𝛿𝑀𝑠 ) of the static stiffness of the machine tool on the workpiece. The sign of 𝛿𝑀𝑠 is always positive and will result in uncut material on the surface.
Fig. 4. Measured cutting force and its quasi-static components in X and Y directions (for a four tooth helical endmill)
𝛿𝐷 = (𝛿𝑇𝑝 + 𝛿𝑇𝑠 ) + (𝛿𝑀𝑝 + 𝛿𝑀𝑠 ) + (𝛿𝑊𝑝 + 𝛿𝑊𝑠 )
(4)
2.2.3. The indirect measurement In the top-down approach an indirect test method schema was planned with the application of LDBB to measure machine tool deviations under loaded and quasi-static conditions. Measurements were taken in one plane of the work volume (called workplane). The workplane was investigated in four different positions, which were in premeditated intersections (see Fig. 5.) for the investigation of the position and direction dependent stiffness.
Fig. 5. The different measurement positions and the corresponding trajectories for the LDBB measurements in the top-down approach
During the measurements the following test parameters were documented: diameter of the nominal path (Ød) feed rate contouring direction (clockwise or counter-clockwise) location of the measuring instrument in the machine tool working zone data capture range (always set to 360°) start and stop points of the measurements actual toolpath in the measurement points (machine axes moved to produce the actual path)
The linearity and hysteresis of the system are important factors to be considered. Repeated measurements on different load levels need to indicate linear load-deflection characteristic to accurately apply the model in the investigated interval. The hysteresis was measured under smaller preload following ISO 230-1 [7]. 2.2.4. The computational model (top-down) The computational model of the top-down approach is applied to describe the variation of the position and direction dependent static stiffness of the machine tool. During the top-down modeling (Fig. 6), the whole system is decomposed into sub-systems. In
this case, the aim is to decompose the indirect measurement data containing superposed errors of parallel motion of more machine axes at the same time, and separate kinematic errors, hysteresis and the quasi-static load induced deviations. Experimental conditions Nominal diameter (ød) Setup in XY-plane (4 pos.)
Different feedrates Different load levels
Unloaded measurement data
Loaded measurement data
Statistical analysis and measurement postprocessing Nominal circular toolpath Actual toolpath in the measurement points
Quasi-static load induced deviations
Hooke s law and stiffness equation
Estimation of position dependent directional stiffness values Fig. 6. The concept of the top-down approach
As can be seen in Fig. 7, the directions and the positions are defined corresponding to the Cartesian machine coordinate system (O) or the spherical measurement coordinate system (T table side and S spindle side).
Fig. 7. Coordinate frames for the three-axis machine tool. (a) before load and (b) after applied load.
The stiffness needs to be characterized in the machine coordinate system. For this: 1. The position dependent directional stiffness values have to be estimated from the measured local point stiffness in the spherical coordinate system of the measurements 2. The spherical coordinates have to be transformed to the corresponding Cartesian coordinate system of the machine tool F( , ) kp (5) ( , ) F( , )
( , )
F sin cos 2 + F sin sin 2 + F cos 2 2
F sin cos F sin sin F cos + + k kx ky z 2
2
(6)
kp
kx k y kz
k z sin cos k y + sin 2 k x + cos 2 k x k y 2
2
2
2
(7)
The simplified equation for the 2 dimensional measurement setup, where θ=0 and the effect of 𝑘𝑧 is neglected: kp
kx ky sin k x + cos 2 k y 2
2
2
(8)
The solution of this equation practically leads to the estimation of the cross compliance at the measurement points. In the corresponding X and Y axis directions (for given φ values) the parameters of the model are calibrated from the point stiffness results. Finally, a linear interpolation was applied to expand the predicted stiffness values to the whole workplane. 3. Implementation of the methodology The introduced methodology was implemented on a three-axis vertical machine tool with C type configuration (Fig. 2). The effective travel range of the axes is 1000 mm for X, 510 mm for Y and 561 mm for Z. Laser interferometer and LDBB measurement results for the bottom-up approach can be found in the appendix. 3.1. The estimation of the bottom-up model After each error source was integrated into the model and the HTMs were constructed to express the corresponding spatial transformation, the “actual“ tool trajectory due to the effect of the geometrical inaccuracies of each axis can be calculated. In Fig. 8 shows the estimated geometric errors from the ‘nominal’ toolpath, which coincides with the third position in the top-down part.
IεxI = 16.8 µm IεyI = 10.4 µm IεzI = 24.4 µm
ød
IεxI = 1.3 µm IεyI = 0.8 µm Iεz I = 38.4 µm
Fig. 8. The estimated (and 1000x magnified) geometric errors in X,Y and Z directions (respectively 𝜀𝑥 , 𝜀𝑦 , 𝜀𝑧 ) around the circular toolpath of one of the LDBB measurement trajectories (Ød=300 mm)
The model enables the assessment of the contributor error factors. The obviously higher errors in Z direction are the result of the pitch error of X axis (with ~70% contribution). 3.2. The estimation of the top-down part Fig. 9 shows the calculated stiffness result from the measured deviations and the applied load. At the intersections of the circular toolpaths, the same points of the investigated workplane have different point stiffness values. This is the result of the differences in the stiffness response of the system to the change in the direction of the applied force. This confirms the matrix structure in Eq. 2. The result deviates between 8.5 N/µm and 10.8 N/µm.
Fig. 9. The static stiffness of the machine tool at four different positions in the investigated workplane
On Fig. 9 it can be seen, that the most important difference is between the directions (X and Y) and the variation according to the position is the most significant in Y directon. According to the root causes of the variations the it can be stated that the configuration of the axes is more important than the topology of the machine tool (C type) as within one circle in given + and directions the variation is not as significant as in case of different positions. An important aspect can be in this variation the distance from the encoders of the ball screw. The variation in Y direction can be more significant as according to the kinematic chain, X axis is sitting on Y. The output of the computational model, populated with the measurement data, is the estimated stiffness in the measurement points. Fig. 10 demonstrates the estimated and the measured point stiffness values after the determination of the directional stiffness values for measurement points where the direction of the applied load does not correspond to the axis directions. The results show a maximum of 2.8% error.
Fig. 10. The estimated (solid lines) and the measured (dashed lines) point stiffness values after the determination of the directional stiffness values
4. Experimental validation An experimental validation was performed to evaluate the predicted variation of the static stiffness. The aim of the validation was to indicate the quasi-static deformations of the machine tool during the process, so 𝛿𝑀𝑠 in Eq. 4. For this reason the same operation (with the same cutting parameters and helical end mill tool) was executed 6 times on aluminum (6082-T6) workpieces close to the investigated plane in the workspace. An inclination angle was selected to avoid the dependency on the sensitive cutting directions only corresponding to the axes directions. The experimental operation (Fig. 11) was composed from two shoulder down-mills of a reference surface, one with finishing and one with roughing conditions. The inclination of the features was designed to optimize the direction and magnitude of the cutting force. For that a force prediction model was applied. The relative influence of the investigated error sources differs for low and high load ranges. While the geometric errors will dominate in low cutting force ranges (finishing conditions, ap = 15 mm, ae = 0.5 mm, working engagement 7.5 mm2), the effect of the variation of the stiffness will be significant in case of higher loads (roughing, ap = 15 mm, ae = 15 mm, corresponding to 75% of the tool diameter, working engagement 150 mm 2). Cutting parameters were kept the same for each operation: feed rate 1500 mm/min, spindle speed 4000min -1, 𝑣𝑐 = 251 𝑚/𝑚𝑖𝑛.
Fig. 11. One of the three test pieces used for validation and the illustration of the machined features (mm): (a) pre-machined surface after slotting, (b) surface after finishing cut (negligible deflection), (c) surface after rough cut (high deflection)
At the end, the finish and rough surfaces form errors were compared to the nominal radial depth of cut with CMM measurements. The milled surfaces were investigated only besides steady-state force characteristics (see Fig. 11 a, b, c). The results can be seen in Table 2 with a significant overlap between the simulated and measured confidence intervals (in case of k=2). Furthermore the tendencies are clear and the result of the simulation also indicates that the sensitive direction for the variation is Y. The systematical variation in Y direction is much more dominant (as can be seen on the measurement results). Similarly, for X direction the tendencies are visible, however they are less significant. Table 2. The results of the validation for the different operations Relative position Operation number
Variation of the errors due to quasi-static deviation* (k=2)
X (mm)
Y (mm)
Measured (µm)
Simulated (µm)
1.
-335.8
0.5
6±2
4±2
2.
-291.6
110.7
11 ± 3
8±2
3.
-172.3
0.6
2±2
1±2
4.
-128.1
110.8
9±3
5±2
5. (ref)
0
0
0±2
0±2
6.
44.2
110.2
7±2
5±2
* between roughed and finished surfaces
During the machining test, the cutting force was measured with a stationary dynamometer (Kistler Type 9265B with clamping plate 9443B, 1kN/µm specified stiffness). The static force components were determined from these measurements (Fig. 4). The process was repeated in different positions with a maximum deviation of 5% in the measured mean forces. Due to the fact that the process parameters are the same, the source of the variation in the deflections can only be the change in the stiffness.
5. Uncertainty analysis The measurement uncertainty through the computational model affects the predictions. Type A (by statistical analysis) and type B (by evaluating further available information) measurement uncertainty assessment were executed according to the guidelines of GUM [18] in case of the bottom-up approach. The main uncertainty sources were considered [19], such as the measurement device uncertainty, the setup uncertainty, the uncertainty due to the misalignment in the measurement, the uncertainty of the environmental variation, the uncertainty of standard angular optics and the uncertainty of measurement of calibrated angular optics. The standard uncertainty values can be found in Table 3. Table 3. The test uncertainty in case of the bottom-up method outlines the reliability of the approach (k=2) X-axis
Y-axis
Z-axis
(0 – 1000 mm)
(0 – 510 mm)
(0 – 561 mm)
Contributors
Test uncertainty
Liner positioning error
11.6 µm
3.8 µm
2.8 µm
Straightness error (Y, X, Y)
4.0 µm
3.4 µm
2.6 µm
Straightness error (Y, Z, X)
3.6 µm
3.2 µm
2.6 µm
Angular Pitch (Tilt I. for Z)
3.2 µm/mm
1.8 µm/mm
1.2 µm/mm
Angular Yaw (Tilt II. for Z)
2.0 µm/mm
2.0 µm/mm
1.2 µm/mm
For the top-down approach the standard uncertainty was estimated after the directions of ISO 230-9 [20] as the maximum value of the estimated standard deviation of the repetitions (Table 4). Variation in case of different positions can be the result of the changes in the setup, as well as the variation of the repeatability of the machine tool. Table 4. The test uncertainty in case of the top-down method outlines the reliability of the approach (k=2) Parameters Standard uncertainty
Pos. 1
Pos. 2
Pos. 3
Pos. 4
2.3 µm
1.6 µm
1.5 µm
1.9 µm
6. Conclusion and discussion A novel methodology is proposed in this paper to predict, under loaded and quasi-static conditions, machine tool errors represented as deviations. The procedure is based on a combination of top-down i.e. indirect measurement of the force – deviation functions at the interface between the tool holder and table, and bottom-up i.e. unloaded direct measurement of individual axis geometric errors. Measurements are combined in a computational model, based on an analytical representation of the machine tool, to compute the aggregated deviation from the nominal positions due to quasi-static forces and geometric errors acting in the machine tool. To validate the proposed methodology predicted deviations were compared with results obtained from machining test. A special test piece was designed to allow the separation of geometric errors from errors due to the variation in stiffness. The results were well confirmed, measured and computed values are overlapping as shown in Table 2. The analysis shows that the deviations, due to variation in static stiffness of the machine tool, can be in the same magnitude as the geometric errors. Comparing the estimated uncertainty for the top-down and bottom-up approaches reveals that the major factor influencing the uncertainty comes from the several direct measurements of the bottom-up approach. The top-down approach - using performance data from indirect measurements may result in lower measurement uncertainties, as it is based on single measurement data gathered at the interface between the toolholder - table. It has long been recognized that machine tools have been considered as individuals, whose properties are not only time and operational dependent, but also design and construction dependent (e.g. how a machine is assembled). The introduction of the proposed concept supports the development of deeper understanding of the causes to this ‘individualism’ by enabling error analysis under loaded conditions. Examples of possible application areas for the presented concept are machine tool manufacturers and users. By applying the proposed concept, machine tool manufacturers can quantify the effect of a design change and/or a variation in assembly of the aggregated system behaviour. This way, optimal system performance can be utilised. Machine tool end-users can use the proposed concept for monitoring changes for prediction of equipment health and prognostic intelligence to significantly enhance asset availability and minimize unscheduled maintenance. Acknowledgement The authors would like to thank Dr. Mikael Hedlind at Scania CV AB for his research support on kinematic modeling. This work was financed through the center for Design and Management of Manufacturing Systems DMMS and supported by XPRES – Initiative for excellence in production research in Sweden.
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Appendix
Fig. 12: The measured component errors of the machine tool in the case study for the bottom-up approach (light and dark color marks the different directions on the axis). The predicted results are indicating the significant contribution of the pitch error of X axis (𝐸𝐵𝑋 ).
Fig. 13: The measured deviations and the pure deflections at each measurement position of the machine tool in the case stud for the top-down approach.
S0141-6359(17)30201-5 http://dx.doi.org/doi:10.1016/j.precisioneng.2017.07.011 PRE 6621
To appear in:
Precision Engineering
Received date: Revised date: Accepted date:
27-4-2017 20-7-2017 25-7-2017
Please cite this article as: Szipka K´aroly, Laspas Theodoros, Archenti Andreas.Measurement and analysis of machine tool errors under quasi-static and loaded conditions.Precision Engineering http://dx.doi.org/10.1016/j.precisioneng.2017.07.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Measurement and analysis of machine tool errors under quasi-static and loaded conditions Károly Szipka*, Theodoros Laspas, Andreas Archenti KTH Royal Institute of Technology, Brinellvägen 68, 10044 Stockholm * Corresponding author. Tel.: +46-736-737-825 . E-mail address: [email protected]
Highlights
A novel methodology for the prediction of machine tool errors under quasi-static and loaded conditions Measurement and analysis of the combined effect of geometric errors and quasi-static load induced deviations on machine tools Estimation of deflections through the determination of the position- and direction-dependent static stiffness of machine tools Introduction how the static stiffness characteristic of machine tools is mapped on the part geometry
Abstract Machine tool testing and accuracy analysis has become increasingly important over the years as it offers machine tool manufacturers and end-users updated information on a machine’s capability. A machine tool´s capability may be determined by mapping the distribution of deformations and their variation range, in the machine tool workspace, under the cumulative effect of thermal and mechanical loads. This paper proposes a novel procedure for the prediction of machine tool errors under quasi-static and loaded conditions. Geometric errors and spatial variation of static stiffness in the work volume of machines are captured and described through the synthesis of bottom-up and top-down model building approaches. The bottom-up approach, determining individual axis errors using direct measurements, is applied to estimate the geometric errors in unloaded condition utilizing homogeneous transformation matrix theory. The top-down approach, capturing aggregated quasi-static deviations using indirect measurements, estimates through an analytical procedure the resultant deviations under loaded conditions. The study introduces a characterization of the position and direction dependent static stiffness and presents the identification how the quasi-static behavior of the machine tool affects the part accuracy. The methodology was implemented in a case study, identifying a variation of up to 27% in the stiffness response of the machine tool. The prediction results were experimentally validated through cutting tests and the uncertainty of the measurements and the applied methodology was investigated to determine the reliability of the predicted errors. Keywords: Machine tool, accuracy, quasi-static stiffness, geometric error
1. Introduction Thorough characterization of machine tools’ accuracy requires quasi-static analysis under loaded and unloaded conditions. Machine tool accuracy analysis can be used to extract valuable information through identification of the separate effects of the various error sources, including kinematic, static, thermal, dynamic, and motion control [1]. For machine tool geometric error analysis, several different schemes have been proposed for ‘direct’ [2] and ‘indirect’ [3] measurements. Direct methods are implemented through error motion measurements of a single axis while indirect methods require simultaneous motion of two or more axis of the machine under test for estimating overall geometric errors. Analysis of superposed errors for a chain of axis uses a ‘‘bottom-up’’ approach, based on Hartenberg–Denavit transformation [4], where motion errors of individual components are added to the aggregated volumetric error [5]. The indirectly measured aggregated errors [6] can be analyzed through a “top-down” approach to derive the deviation matrix of the system. ISO 230-1 [7] specifies methods and guidelines for testing the accuracy of machine tools under unloaded or quasi-static conditions accounting only for geometric errors and quasi-static load induced deviations. Under machining conditions, however,
the induced deviations due to loads have a dependency on the machine tool stiffness and influence the system’s precision. Machine tool stiffness represents its ability to resist deflections due to e.g. cutting forces and have a significant effect on the quasistatic and dynamic accuracy. Load induced deformations have been commonly described and investigated by various types of finite element (FE) approaches [8] or combination of FE and analytical methods [9] rather than measurement based modeling approaches [10], [11]. [12] and in dynamic case [13] presents direction dependent stiffness investigations. The main limitation with the existing methods is the lack of appropriate measurement approaches that link individual axis errors under the influence of loads to the aggregated machine tool accuracy. The reasons can partly be explained by the lack of proper instrumentation, which is usually limited to the measurement of displacement in one position and direction for one setup [7], neglecting the characterization of the machine tool`s response in the whole workspace.
Fig. 1. The synthesis of bottom-up and top down modeling approach
The proposed methodology relies on the synthesis of bottom-up (geometric errors stack-up) and top-down (position dependent variation of static stiffness) approaches and enables the relaying of the investigated error sources through the combination of indirect and direct measurements (Fig. 1). Hence, the two approaches are acting complementary to each other covering different aspects of the machine tool performance with enabling the possibility for assessment of the pure machine tool deflections without the geometric errors. The core novelty of this work relies on the top-down approach and the synthesis of the two approaches. The bottom-up approach aims in evaluating the geometric accuracy of the machine tool under unloaded conditions at the trajectory of the functional point (as described in ISO 230-1 [7]). Using homogeneous transformation matrices (HTM) the geometric accuracy of the machine tool is determined by superposing the geometric errors of each individual machine axis. Direct measurement of the errors can be carried out using a laser interferometer or other standard measurement methods [2]. The top-down approach aims at mapping the static stiffness response of the machine tool by measuring the aggregated deviations under loaded conditions and hence allows the integration of small machine structure deformations in the geometric error model. In addition, it enables the identification of dependencies of stiffness variation in the workspace due to axes positions and applied force direction. An indirect measurement procedure for measuring the aggregated errors is realized experimentally by the test instrument Loaded Double Ball Bar (LDBB) [14], which is capable of applying forces of controlled magnitude and orientation and measure the induced deviations identifying the stiffness variation under different machine spatial configurations. The LDBB is seen as an elastic link – as described by the Elastic Linked System (ELS) [15] framework – that closes the force loop between the tool and the workpiece and represents the static response of the machine structure to the cutting process. In section 2 of the paper an introduction of the methodology for the assessment of geometric and quasi-static load dependent errors is provided with a focus on the top-down approach. Section 3 covers the implementation of the proposed methodology, with additional measurement results in the appendix, and Section 4 presents the experimental validation procedure. Section 5 focuses on the measurement uncertainties of both of the approaches before conclusions are discussed in Section 6. 2. Methodology for the assessment of geometric and quasi-static load dependent errors 2.1. The bottom-up approach Characterization of individual axis position-dependent and position-independent geometric errors (e.g., linear positioning, straightness, angular or squareness deviation) under unloaded conditions requires a direct measurement method. Laser interferometer systems are commonly used for the task [2] as they enable measurement with high precision and low uncertainty as compared to other methods. In this study a Renishaw XL-80 laser interferometer was used. Since all of the modeling methods which follow the principle of the bottom-up modeling approach (axis geometric error stack-up) are sensitive to the uncertainty of the contributing error components, the uncertainty of the measurements is of importance and will be further analyzed. For a three-axis machine tool 18 position dependent and 3 position independent (location) errors can be identified. Since all zero positions of linear axes can be set to zero when checking the geometric accuracy, any possible offset errors can be neglected. The location errors of the spindle are simplified, which means 4 location errors less to be considered. By selecting Z axis as primary and Y and X as secondary and tertiary axis, the location errors will define the location and orientation with respect to the average line of Z axis. The considered component and location errors can be found in Table 1 using the notation of ISO 230-1 [7]. Table 1. The component and location errors of a three-axis machine tool
Linear errors Axis
Positioning
X axis
EXX
EYX
Y axis
EYY EZZ
EXY EYZ
Z axis Location Errors
Angular errors
Straightness
Roll
Pitch
Yaw
EZX
EAX
EBX
ECX
EZY EXZ
EBY ECZ
EAX EAZ
ECY EBZ
𝐸𝐴0𝑌 squareness error of Y to Z; 𝐸𝐶0𝑋 squareness error of X to Y; 𝐸𝐵0𝑋 squareness error of X to Z;
2.1.1. The computational model (bottom-up) The bottom-up approach is based on HTM theory to describe the kinematic structure of the machine. In order to achieve a comprehensive model, certain steps are defined. A systematic procedure can be outlined through the following steps: 1. Identification of the kinematic structure of the machine model 2. Assignment of the global reference frame and axis local coordinate frames to facilitate the modeling procedure 3. Modeling of the nominal kinematic configuration by describing the relative position and orientation of the axes with respect to each other using HTMs 4. Definition of the measurement functional point where individual error components have been measured 5. Construction of the geometric error model and population with measured data The kinematic structure sets the base on which the entire modeling process will rely, in order to express the motion of the rigid axes and the linking joints. Local and global coordinate frames have to be chosen in order to facilitate the integration of the measurement data with the model. The global coordinate frame is defined to meet the corresponding 0 coordinate for each axis for a three axis machine. An example can be seen in Fig. 2. The model is composed to be applicable for orthogonal axis machine tool with maximum three translational axes. The position independent location errors and the position dependent component errors for each axis give the considered error parameters (see Table 1), which are measured separately. The measured error parameters are coupled into HTMs to express the deviation of each individual axis from the nominal position, the effect on the succeeding rigid body frames and the effect on the accuracy in the position between the tool and workpiece. The HTM theory is implemented for geometric error modeling according to [5] and [16], using the equations defined by these two papers.
Fig. 2. The assignment of the coordinate frames for the three-axis machine tool (AFM R-1000). The coordinate system (O) corresponds to the reference coordinate frame, and coordinate systems (F) correspond to the local coordinate frames of the X, Y and Z axis.
At the end of the process the position and the orientation of the functional point is determined in respect to the table. The relationship between the functional point and the measurement points is a sensitive part of the model as measurements are implemented in one track of the workspace and conclusion is made in another trajectory. To avoid redundancies the lever effect of the angular errors due to the offset between the functional point and the measurement point was compensated. The summarized concept of the bottom-up approach can be seen in Fig. 3.
Machine Tool kinematic structure
Measurement data
Defined toolpath
Statistical analysis
Individual axes position
Geometric error model Composition of HTMs Offset between the measurement and the functional point
Linear and rotational motion carriage with errors of single axes
Estimated geometric errors for specified toolpath Fig. 3. The concept of the bottom-up approach
2.2. The top-down approach The top-down approach is the detailed description of the effect of quasi-static forces, which are related to the static stiffness of the machine tool. In this approach, a computational model is developed to analyze forces and deflections based on an indirect measurement schema, which allows measuring superposed errors of simultaneous motion of two or more machine axes. Investigations were made in one certain plane of the machine tools’ work volume. 2.2.1. Static stiffness of machine tools The stiffness of machine tools generally expresses the capacity of the machine to endure loads within its workspace. The resultant stiffness (aggregated) of a machine tool (at a certain location and direction) is defined through the ratio of the applied load and the induced deformation between the spindle nose and the machine tool table. The loading process has to be slow (quasistatic condition) in case of the static stiffness. For periodic loading this means 1 or 0.5 Hz or lower [8], [17], according to the damping of the mechanical system of the machine tool, to avoid transient movements. The magnitude of the elastic deformations depends on the stiffness of all the components of the machine’s structural chain. In general, it can be stated that the weakest link, the component where the stiffness is the lowest in the stiffness chain, will dominate the response of the whole machine. This means a high level of sensitivity and highlights the importance of stiffness response characterization. Most commonly, the static stiffness of a machine tool is determined through deflection measurements in the directions of the axes, where deflection is measured in line with the applied force. ISO 230-1 [7] introduce the guidelines for the test of machine static compliance at a certain position in the working volume. This method gives an impression about the stiffness state of the machine in the axis direction, but can’t provide enough information for an adequate characterization or an effective compensation. Such measurements can determine a local point stiffness value (kp), which is the magnitude of the applied force (F) divided by the displacement (δ) between the spindle nose with respect to the machine tool table in the load direction: 𝑘𝑝 =
𝐹
𝛿
(1)
Regarding the changing relative position and/or orientation between the mechanical components during motion, the resultant deflections will differ in case of different directions (𝑒𝑓 ) of the force and different positions (p) at the workspace. Consequently the static stiffness of the machine, K p, e f varies as the function of the position and the direction (signum function) of the applied force:
k xx (p,sgn(e f, x )) ... ... K (p,e f ) ... k yy (p,sgn(e f, y )) ... ... ... k zz (p,sgn(e f, z ))
(2)
It has to be noted that certain deflections are the result of internal, assembly dependent loads of the machine, including the effect of the weight of the machine and the preloading of components. Even though these parameters are included in the presented model through the measurements, their change can affect the result. 2.2.2. The mapping of static stiffness characteristic on the part form accuracy It is known that the static stiffness of machine tools varies, both regarding the behavior of each individual machine as well as between different machine tool configurations [15]. This variation, if not compensated, will be mapped on the form accuracy of the part as deviations due to loads. Thus, the static stiffness characteristic of a machine tool is mapped on the form accuracy of the part. The total deviation D (nominal surface) can be decomposed into three different components: deflection of the tool (𝛿 T), deformation of the machine, including fixture and toolholder (𝛿 M), and the deformation of the workpiece (𝛿 W).
𝛿𝐷 = 𝛿𝑇 + 𝛿𝑀 + 𝛿𝑊
(3)
These deviations are the response of the system to the process force. Two characteristics can be identified in the system response: the pulsating component (index p) and a static component (index s). The separation of these two components leads to Eq. 4. A typical example of a measured force in steady-state can be seen in Fig. 4. The static component is responsible for the direct mapping (𝛿𝑀𝑠 ) of the static stiffness of the machine tool on the workpiece. The sign of 𝛿𝑀𝑠 is always positive and will result in uncut material on the surface.
Fig. 4. Measured cutting force and its quasi-static components in X and Y directions (for a four tooth helical endmill)
𝛿𝐷 = (𝛿𝑇𝑝 + 𝛿𝑇𝑠 ) + (𝛿𝑀𝑝 + 𝛿𝑀𝑠 ) + (𝛿𝑊𝑝 + 𝛿𝑊𝑠 )
(4)
2.2.3. The indirect measurement In the top-down approach an indirect test method schema was planned with the application of LDBB to measure machine tool deviations under loaded and quasi-static conditions. Measurements were taken in one plane of the work volume (called workplane). The workplane was investigated in four different positions, which were in premeditated intersections (see Fig. 5.) for the investigation of the position and direction dependent stiffness.
Fig. 5. The different measurement positions and the corresponding trajectories for the LDBB measurements in the top-down approach
During the measurements the following test parameters were documented: diameter of the nominal path (Ød) feed rate contouring direction (clockwise or counter-clockwise) location of the measuring instrument in the machine tool working zone data capture range (always set to 360°) start and stop points of the measurements actual toolpath in the measurement points (machine axes moved to produce the actual path)
The linearity and hysteresis of the system are important factors to be considered. Repeated measurements on different load levels need to indicate linear load-deflection characteristic to accurately apply the model in the investigated interval. The hysteresis was measured under smaller preload following ISO 230-1 [7]. 2.2.4. The computational model (top-down) The computational model of the top-down approach is applied to describe the variation of the position and direction dependent static stiffness of the machine tool. During the top-down modeling (Fig. 6), the whole system is decomposed into sub-systems. In
this case, the aim is to decompose the indirect measurement data containing superposed errors of parallel motion of more machine axes at the same time, and separate kinematic errors, hysteresis and the quasi-static load induced deviations. Experimental conditions Nominal diameter (ød) Setup in XY-plane (4 pos.)
Different feedrates Different load levels
Unloaded measurement data
Loaded measurement data
Statistical analysis and measurement postprocessing Nominal circular toolpath Actual toolpath in the measurement points
Quasi-static load induced deviations
Hooke s law and stiffness equation
Estimation of position dependent directional stiffness values Fig. 6. The concept of the top-down approach
As can be seen in Fig. 7, the directions and the positions are defined corresponding to the Cartesian machine coordinate system (O) or the spherical measurement coordinate system (T table side and S spindle side).
Fig. 7. Coordinate frames for the three-axis machine tool. (a) before load and (b) after applied load.
The stiffness needs to be characterized in the machine coordinate system. For this: 1. The position dependent directional stiffness values have to be estimated from the measured local point stiffness in the spherical coordinate system of the measurements 2. The spherical coordinates have to be transformed to the corresponding Cartesian coordinate system of the machine tool F( , ) kp (5) ( , ) F( , )
( , )
F sin cos 2 + F sin sin 2 + F cos 2 2
F sin cos F sin sin F cos + + k kx ky z 2
2
(6)
kp
kx k y kz
k z sin cos k y + sin 2 k x + cos 2 k x k y 2
2
2
2
(7)
The simplified equation for the 2 dimensional measurement setup, where θ=0 and the effect of 𝑘𝑧 is neglected: kp
kx ky sin k x + cos 2 k y 2
2
2
(8)
The solution of this equation practically leads to the estimation of the cross compliance at the measurement points. In the corresponding X and Y axis directions (for given φ values) the parameters of the model are calibrated from the point stiffness results. Finally, a linear interpolation was applied to expand the predicted stiffness values to the whole workplane. 3. Implementation of the methodology The introduced methodology was implemented on a three-axis vertical machine tool with C type configuration (Fig. 2). The effective travel range of the axes is 1000 mm for X, 510 mm for Y and 561 mm for Z. Laser interferometer and LDBB measurement results for the bottom-up approach can be found in the appendix. 3.1. The estimation of the bottom-up model After each error source was integrated into the model and the HTMs were constructed to express the corresponding spatial transformation, the “actual“ tool trajectory due to the effect of the geometrical inaccuracies of each axis can be calculated. In Fig. 8 shows the estimated geometric errors from the ‘nominal’ toolpath, which coincides with the third position in the top-down part.
IεxI = 16.8 µm IεyI = 10.4 µm IεzI = 24.4 µm
ød
IεxI = 1.3 µm IεyI = 0.8 µm Iεz I = 38.4 µm
Fig. 8. The estimated (and 1000x magnified) geometric errors in X,Y and Z directions (respectively 𝜀𝑥 , 𝜀𝑦 , 𝜀𝑧 ) around the circular toolpath of one of the LDBB measurement trajectories (Ød=300 mm)
The model enables the assessment of the contributor error factors. The obviously higher errors in Z direction are the result of the pitch error of X axis (with ~70% contribution). 3.2. The estimation of the top-down part Fig. 9 shows the calculated stiffness result from the measured deviations and the applied load. At the intersections of the circular toolpaths, the same points of the investigated workplane have different point stiffness values. This is the result of the differences in the stiffness response of the system to the change in the direction of the applied force. This confirms the matrix structure in Eq. 2. The result deviates between 8.5 N/µm and 10.8 N/µm.
Fig. 9. The static stiffness of the machine tool at four different positions in the investigated workplane
On Fig. 9 it can be seen, that the most important difference is between the directions (X and Y) and the variation according to the position is the most significant in Y directon. According to the root causes of the variations the it can be stated that the configuration of the axes is more important than the topology of the machine tool (C type) as within one circle in given + and directions the variation is not as significant as in case of different positions. An important aspect can be in this variation the distance from the encoders of the ball screw. The variation in Y direction can be more significant as according to the kinematic chain, X axis is sitting on Y. The output of the computational model, populated with the measurement data, is the estimated stiffness in the measurement points. Fig. 10 demonstrates the estimated and the measured point stiffness values after the determination of the directional stiffness values for measurement points where the direction of the applied load does not correspond to the axis directions. The results show a maximum of 2.8% error.
Fig. 10. The estimated (solid lines) and the measured (dashed lines) point stiffness values after the determination of the directional stiffness values
4. Experimental validation An experimental validation was performed to evaluate the predicted variation of the static stiffness. The aim of the validation was to indicate the quasi-static deformations of the machine tool during the process, so 𝛿𝑀𝑠 in Eq. 4. For this reason the same operation (with the same cutting parameters and helical end mill tool) was executed 6 times on aluminum (6082-T6) workpieces close to the investigated plane in the workspace. An inclination angle was selected to avoid the dependency on the sensitive cutting directions only corresponding to the axes directions. The experimental operation (Fig. 11) was composed from two shoulder down-mills of a reference surface, one with finishing and one with roughing conditions. The inclination of the features was designed to optimize the direction and magnitude of the cutting force. For that a force prediction model was applied. The relative influence of the investigated error sources differs for low and high load ranges. While the geometric errors will dominate in low cutting force ranges (finishing conditions, ap = 15 mm, ae = 0.5 mm, working engagement 7.5 mm2), the effect of the variation of the stiffness will be significant in case of higher loads (roughing, ap = 15 mm, ae = 15 mm, corresponding to 75% of the tool diameter, working engagement 150 mm 2). Cutting parameters were kept the same for each operation: feed rate 1500 mm/min, spindle speed 4000min -1, 𝑣𝑐 = 251 𝑚/𝑚𝑖𝑛.
Fig. 11. One of the three test pieces used for validation and the illustration of the machined features (mm): (a) pre-machined surface after slotting, (b) surface after finishing cut (negligible deflection), (c) surface after rough cut (high deflection)
At the end, the finish and rough surfaces form errors were compared to the nominal radial depth of cut with CMM measurements. The milled surfaces were investigated only besides steady-state force characteristics (see Fig. 11 a, b, c). The results can be seen in Table 2 with a significant overlap between the simulated and measured confidence intervals (in case of k=2). Furthermore the tendencies are clear and the result of the simulation also indicates that the sensitive direction for the variation is Y. The systematical variation in Y direction is much more dominant (as can be seen on the measurement results). Similarly, for X direction the tendencies are visible, however they are less significant. Table 2. The results of the validation for the different operations Relative position Operation number
Variation of the errors due to quasi-static deviation* (k=2)
X (mm)
Y (mm)
Measured (µm)
Simulated (µm)
1.
-335.8
0.5
6±2
4±2
2.
-291.6
110.7
11 ± 3
8±2
3.
-172.3
0.6
2±2
1±2
4.
-128.1
110.8
9±3
5±2
5. (ref)
0
0
0±2
0±2
6.
44.2
110.2
7±2
5±2
* between roughed and finished surfaces
During the machining test, the cutting force was measured with a stationary dynamometer (Kistler Type 9265B with clamping plate 9443B, 1kN/µm specified stiffness). The static force components were determined from these measurements (Fig. 4). The process was repeated in different positions with a maximum deviation of 5% in the measured mean forces. Due to the fact that the process parameters are the same, the source of the variation in the deflections can only be the change in the stiffness.
5. Uncertainty analysis The measurement uncertainty through the computational model affects the predictions. Type A (by statistical analysis) and type B (by evaluating further available information) measurement uncertainty assessment were executed according to the guidelines of GUM [18] in case of the bottom-up approach. The main uncertainty sources were considered [19], such as the measurement device uncertainty, the setup uncertainty, the uncertainty due to the misalignment in the measurement, the uncertainty of the environmental variation, the uncertainty of standard angular optics and the uncertainty of measurement of calibrated angular optics. The standard uncertainty values can be found in Table 3. Table 3. The test uncertainty in case of the bottom-up method outlines the reliability of the approach (k=2) X-axis
Y-axis
Z-axis
(0 – 1000 mm)
(0 – 510 mm)
(0 – 561 mm)
Contributors
Test uncertainty
Liner positioning error
11.6 µm
3.8 µm
2.8 µm
Straightness error (Y, X, Y)
4.0 µm
3.4 µm
2.6 µm
Straightness error (Y, Z, X)
3.6 µm
3.2 µm
2.6 µm
Angular Pitch (Tilt I. for Z)
3.2 µm/mm
1.8 µm/mm
1.2 µm/mm
Angular Yaw (Tilt II. for Z)
2.0 µm/mm
2.0 µm/mm
1.2 µm/mm
For the top-down approach the standard uncertainty was estimated after the directions of ISO 230-9 [20] as the maximum value of the estimated standard deviation of the repetitions (Table 4). Variation in case of different positions can be the result of the changes in the setup, as well as the variation of the repeatability of the machine tool. Table 4. The test uncertainty in case of the top-down method outlines the reliability of the approach (k=2) Parameters Standard uncertainty
Pos. 1
Pos. 2
Pos. 3
Pos. 4
2.3 µm
1.6 µm
1.5 µm
1.9 µm
6. Conclusion and discussion A novel methodology is proposed in this paper to predict, under loaded and quasi-static conditions, machine tool errors represented as deviations. The procedure is based on a combination of top-down i.e. indirect measurement of the force – deviation functions at the interface between the tool holder and table, and bottom-up i.e. unloaded direct measurement of individual axis geometric errors. Measurements are combined in a computational model, based on an analytical representation of the machine tool, to compute the aggregated deviation from the nominal positions due to quasi-static forces and geometric errors acting in the machine tool. To validate the proposed methodology predicted deviations were compared with results obtained from machining test. A special test piece was designed to allow the separation of geometric errors from errors due to the variation in stiffness. The results were well confirmed, measured and computed values are overlapping as shown in Table 2. The analysis shows that the deviations, due to variation in static stiffness of the machine tool, can be in the same magnitude as the geometric errors. Comparing the estimated uncertainty for the top-down and bottom-up approaches reveals that the major factor influencing the uncertainty comes from the several direct measurements of the bottom-up approach. The top-down approach - using performance data from indirect measurements may result in lower measurement uncertainties, as it is based on single measurement data gathered at the interface between the toolholder - table. It has long been recognized that machine tools have been considered as individuals, whose properties are not only time and operational dependent, but also design and construction dependent (e.g. how a machine is assembled). The introduction of the proposed concept supports the development of deeper understanding of the causes to this ‘individualism’ by enabling error analysis under loaded conditions. Examples of possible application areas for the presented concept are machine tool manufacturers and users. By applying the proposed concept, machine tool manufacturers can quantify the effect of a design change and/or a variation in assembly of the aggregated system behaviour. This way, optimal system performance can be utilised. Machine tool end-users can use the proposed concept for monitoring changes for prediction of equipment health and prognostic intelligence to significantly enhance asset availability and minimize unscheduled maintenance. Acknowledgement The authors would like to thank Dr. Mikael Hedlind at Scania CV AB for his research support on kinematic modeling. This work was financed through the center for Design and Management of Manufacturing Systems DMMS and supported by XPRES – Initiative for excellence in production research in Sweden.
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Appendix
Fig. 12: The measured component errors of the machine tool in the case study for the bottom-up approach (light and dark color marks the different directions on the axis). The predicted results are indicating the significant contribution of the pitch error of X axis (𝐸𝐵𝑋 ).
Fig. 13: The measured deviations and the pure deflections at each measurement position of the machine tool in the case stud for the top-down approach.